An arbitrary-precision number represented using scientific notation.

This type describes the set of all Reals which have a finite decimal expansion.

A scientific number with coefficient c and base10Exponent e corresponds to the Fractional number: fromInteger c * 10 ^^ e

scientific c e constructs a scientific number which corresponds to the Fractional number: fromInteger c * 10 ^^ e.

The coefficient of a scientific number.

Note that this number is not necessarily normalized, i.e. it could contain trailing zeros.

Scientific numbers are automatically normalized when pretty printed or in toDecimalDigits.

Use normalize to do manual normalization.

The base-10 exponent of a scientific number.

Return True if the scientific is a floating point, False otherwise.

Also see: floatingOrInteger.

Return True if the scientific is an integer, False otherwise.

Also see: floatingOrInteger.

Like fromRational, this function converts a Rational to a Scientific but instead of diverging (i.e loop and consume all space) on repeating decimals it detects the repeating part, the repetend, and returns where it starts.

To detect the repetition this function consumes space linear in the number of digits in the resulting scientific. In order to bound the space usage an optional limit can be specified. If the number of digits reaches this limit Left (s, r) will be returned. Here s is the Scientific constructed so far and r is the remaining Rational. toRational s + r yields the original Rational

If the limit is not reached or no limit was specified Right (s, mbRepetendIx) will be returned. Here s is the Scientific without any repetition and mbRepetendIx specifies if and where in the fractional part the repetend begins.

For example:

fromRationalRepetend Nothing (1 % 28) == Right (3.571428e-2, Just 2)

This represents the repeating decimal: 0.03571428571428571428... which is sometimes also unambiguously denoted as 0.03(571428). Here the repetend is enclosed in parentheses and starts at the 3rd digit (index 2) in the fractional part. Specifying a limit results in the following:

fromRationalRepetend (Just 4) (1 % 28) == Left (3.5e-2, 1 % 1400)

You can expect the following property to hold.

 forall (mbLimit :: Maybe Int) (r :: Rational).
 r == (case fromRationalRepetend mbLimit r of
        Left (s, r') -> toRational s + r'
        Right (s, mbRepetendIx) ->
          case mbRepetendIx of
            Nothing         -> toRational s
            Just repetendIx -> toRationalRepetend s repetendIx)

Converts a Scientific with a repetend (a repeating part in the fraction), which starts at the given index, into its corresponding Rational.

For example to convert the repeating decimal 0.03(571428) you would use: toRationalRepetend 0.03571428 2 == 1 % 28

Preconditions for toRationalRepetend s r:

• r >= 0

• r < -(base10Exponent s)

The formula to convert the Scientific s with a repetend starting at index r is described in the paper: turning_repeating_decimals_into_fractions.pdf and is defined as follows:

  (fromInteger nonRepetend + repetend % nines) /
  fromInteger (10^^r)
where
  c  = coefficient s
  e  = base10Exponent s

  -- Size of the fractional part.
  f = (-e)

  -- Size of the repetend.
  n = f - r

  m = 10^^n

  (nonRepetend, repetend) = c `quotRem` m

  nines = m - 1

Also see: fromRationalRepetend.

floatingOrInteger determines if the scientific is floating point or integer. In case it's floating-point the scientific is converted to the desired RealFloat using toRealFloat.

Also see: isFloating or isInteger.

Safely convert a Scientific number into a RealFloat (like a Double or a Float).

Note that this function uses realToFrac (fromRational . toRational) internally but it guards against computing huge Integer magnitudes (10^e) that could fill up all space and crash your program. If the base10Exponent of the given Scientific is too big or too small to be represented in the target type, Infinity or 0 will be returned respectively. Use toBoundedRealFloat which explicitly handles this case by returning Left.

Always prefer toRealFloat over realToFrac when converting from scientific numbers coming from an untrusted source.

Preciser version of toRealFloat. If the base10Exponent of the given Scientific is too big or too small to be represented in the target type, Infinity or 0 will be returned as Left.

Convert a Scientific to a bounded integer.

If the given Scientific doesn't fit in the target representation, it will return Nothing.

This function also guards against computing huge Integer magnitudes (10^e) that could fill up all space and crash your program.

Convert a RealFloat (like a Double or Float) into a Scientific number.

Note that this function uses floatToDigits to compute the digits and exponent of the RealFloat number. Be aware that the algorithm used in floatToDigits doesn't work as expected for some numbers, e.g. as the Double 1e23 is converted to 9.9999999999999991611392e22, and that value is shown as 9.999999999999999e22 rather than the shorter 1e23; the algorithm doesn't take the rounding direction for values exactly half-way between two adjacent representable values into account, so if you have a value with a short decimal representation exactly half-way between two adjacent representable values, like 5^23*2^e for e close to 23, the algorithm doesn't know in which direction the short decimal representation would be rounded and computes more digits

Like show but provides rendering options.

Control the rendering of floating point numbers.

Similar to floatToDigits, toDecimalDigits takes a positive Scientific number, and returns a list of digits and a base-10 exponent. In particular, if x>=0, and

toDecimalDigits x = ([d1,d2,...,dn], e)

then

1. n >= 1

2. x = 0.d1d2...dn * (10^^e)

3. 0 <= di <= 9

4. null $ takeWhile (==0) $ reverse [d1,d2,...,dn]

The last property means that the coefficient will be normalized, i.e. doesn't contain trailing zeros.

Normalize a scientific number by dividing out powers of 10 from the coefficient and incrementing the base10Exponent each time.

You should rarely have a need for this function since scientific numbers are automatically normalized when pretty-printed and in toDecimalDigits.